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We study linear series on a general curve of genus $g$, whose images areexceptional with regard to their secant planes. Working in the framework of anextension of Brill-Noether theory to pairs of linear series, we prove that ageneral curve has no linear series with exceptional secant planes, in a veryprecise sense, whenever the total number of series is finite. Next, wepartially solve the problem of computing the number of linear series withexceptional secant planes in a one-parameter family in terms of tautologicalclasses associated with the family, by evaluating our hypothetical formulaalong judiciously-chosen test families. As an application, we compute thenumber of linear series with exceptional secant planes on a general curveequipped with a one-dimensional family of linear series. We pay specialattention to the extremal case of $d$-secant $(d-2)$-planes to$(2d-1)$-dimensional series, which appears in the study of Hilbert schemes ofpoints on surfaces. In that case, our formula may be rewritten in terms ofhypergeometric series, which allows us both to prove that it is nonzero and todeduce its asymptotics in $d$.

Geometry of curves with exceptional secant planes: linear series along the general curve